Leeds General Infirmary has been given the go-ahead to resume children's heart surgery. Photograph: Lynne Cameron/PA

Children's heart surgery is to be phased in again slowly over the next month at the Leeds General Infirmary, where it was suspended over concerns about death rates, the quality of surgery and staffing levels.

But NHS England, announcing that low-risk operations could begin again on Wednesday, said that improvement in a number of areas would be required and it would continue to explore issues raised about the unit.

A review team worked through the weekend to assess whether the children's heart surgery unit was safe. At the end of what NHS England described as the first stage of a review, it advised that the quality of surgery and staffing were sufficient to allow a phased resumption of operations.

NHS England said the review "found that the [Leeds Teaching Hospitals NHS Trust] data for monitoring surgical results was uniquely poor, triggering concerns about death rates and gaps in information.

The hospital had not been submitting all the required information about the surgical procedures it was carrying out on children, the Guardian understands. The review wants "significant improvements to the way the unit monitors the quality of care" so the outcomes at Leeds can be properly compared with the outcomes at other hospitals.

The second stage of the review will look at some of the other issues that have been raised including "the way complaints from patients are handled, including the issues raised by the Children's Heart Federation" – an umbrella organisation for a number of patient support groups. A review of patients' case notes from the past three years will be completed.

Sir Bruce Keogh, medical director of NHS England, said: "The information that came to light about Leeds raised some really serious questions and action had to be taken. The trust agreed to pause surgery until these questions were investigated.

"If we have learned anything from public inquiries such as Bristol and Mid Staffordshire, it is that patients were harmed while organisations argued about the veracity of data used to measure clinical results, rather than addressing the underlying issues. We would not have been forgiven if a child had died or suffered unnecessary harm while we sat on our hands.

"I am pleased that we have now been given assurances by independent assessors that the immediate safety concerns, which were bubbling up from a variety of sources, have been addressed and that the unit can recommence surgery.

"We now need to explore some of the wider issues around how the unit operates as a whole. I hope we will soon be able to give the unit a full clean bill of health beyond this immediate reassurance of safety."

Keogh said the proper collection of data about patients' treatment was vital so it could be clearly seen how well hospitals were performing. "I want to be clear that NHS England will do everything in its power to make sure that measuring clinical outcomes will be given priority in the new NHS. Organisations cannot know they are providing effective or safe care unless they are measuring and monitoring their services," he said.

Campaigners who have been fighting to prevent the closure of the Leeds unit as part of an NHS reorganisation said they were relieved at the news but still had suspicions about the events that led to the suspension of the service a day after they had won a judicial review in the high court.

Save Our Surgery said: "Our main focus since the suspension of the service has been on ensuring surgery resumes as quickly as possible. However, we do have numerous concerns about the chain of events that led to this decision.

"We seek urgent reassurance from the secretary of state for health that actions taken were in no way related to our high court judgement the previous day, and we call for an investigation into how unverified mortality data and 'rumblings' from parties with an interest in the outcome of the judicial review could have led to the unit being shut so dramatically."